Critical dynamics of capillary waves in an ionic liquid: XPCS studies

Eli Sloutskin

Physics DepartmentBar-Ilan University

IsraelNow at SEAS, Harvard

OUTLINE

Introduction & motivation:

Ionic liquids and their surface structure.

Experimental technique: surface XPCS.

Spontaneous heterodyne - homodyne switching.

Results:

Propagating and overdamped regimes of thermal CW.

Critical behavior of thermal CW excitations.

Future directions:

Can XPCS measure surface viscoelasticity ?

Room Temperature Molten SaltsClassical Salts:

NaCl Tm= 1074 K

MgCl2 Tm= 1260 K

YBr3 Tm= 1450 K

RT = 101 cPTm = -71 oC

Butylmethylimidazoliumtetrafluoroborate

Ionic liquids: Solvent free electrolytes. “Green” industry. 100s synthesized since ’97.

The static surface structure of ILsX-ray reflectivity studies

Surface induced ordering is required to fit the reflectivity.

Sloutskin et al., JACS 127, 7796 (2005)

Molecular dynamics

Oscillatory surface electron density profile Surface mixture of cations and anions

R.M. Lynden-Bell and M. Del Pópolo, PCCP 8, 949 (2006).

B.L. Bhargava and S. Balasubramanian, JACS 128, 10073 (2006).

Sloutskin et al., J. Chem. Phys. 125, 174715 (2006).

-40 -20 00.0

0.4

0.8

1.2

z(Å)

(e

/Å3 )

Dynamic light scattering

Modified elastic term in CW spectrum. Possible ferroelectric order-disorder phase transition at T≈380 K.

V. Halka, R. Tsekov, and W. Freyland, PCCP 7, 2038 (2005).

XPCS yields the S(q,) of T-excited surface capillary waves. No contribution from bulk modes. Higher k-vectors can be probed. To date no XPCS for any ionic liquid.

The surface dynamics of ILs

Tc (???)

Su

rfac

e d

ipo

le d

ensi

ty

X-ray photocorrelation spectroscopy (XPCS)z

o

I

qx

qz

qx

qkin

kout

< c

A.

Ma

dse

n e

t a

l., w

eb

site

of

ES

RF

.

qx

Intensity autocorrelation G() at a given wave vector qx is related to the dynamics of surface excitations for the same wave vector.

Propagating mode

t

= Specular reflection

10-6 10-5 10-4 10-3

(sec)

≠ ≠ Diffuse scattering by Diffuse scattering by CWCW

10-5 10-4 10-3

(sec)

G()

Overdamped mode

t G

)

Troïka (ESRF)

(sec)

G

)

Capillary wave excitations with shorter wavelengths have higher frequencies.

Damping increases for shorter wave lengths. Viscous dissipation is due to velocity gradients: ∆v term in the Navier-Stokes equation.

At high qx and low T, capillary waves are overdamped.

Experimental G(qx,t) functions: qualitative description.

=ck

k

10-6 10-5 10-4 10-3

qx=78 mm-1

qx=56 mm-1

qx=36 mm-1

qx=26 mm-1

qx=13 mm-1

(s)

T=343 K

G(q

x,)

10-5 10-4 10-3

T=407 K

T=353 K

T=333 K

T=323 K

T=313 K

(s)

G(q

x,)

overdampedT=40 oC

qx=24 mm-1PropagatingT=134 oC

Theoretical descriptionFor incompressible fluid: divv = 0

Assume: liquid density and viscosity remain constant up to the dividing surface

E.H. Lucassen-Reynders and J. Lucassen, Adv. Colloid Interface Sci. 2, 347 (1969)

0 xz

i

k

k

iikik x

v

x

vp

02

2

zzx

h

Boundary conditions at the interface:Stress tensor:

Surface tension

(ii)

(i) ideal interface

Young-Laplace:

21

11

RRp

hx

z

Navier-Stokes: zgvpvvt

vˆ

0~, kDDispersion relation for the capillary waves:

i~temporal damping

Transition from propagating to overdamped waves: hydrodynamical theory

Tc=35o C @ k=24 mm-1

Solve y(Tc)= 0.145

Experimental Tc is higher than 40 oC !!!

145.04 2

k

y

Dispersion relation

Damping

Propagating

log(y)

log(

), lo

g(

)

Use independently-measured (T), (T) and (T)

“- Byrne and Earnshaw (1979)

“~

10-5 10-4 10-3

T=40 oC

(sec)

G(q

x,) k=24 mm-1

XPCS measures the actual population of ripplon energy levels at a given T, not the energy levels allowed by hydrodynamics.

Linear response theory (Jäckle & Kawasaki, 1995)

Spectrum of surface ripplons:

J. Jäckle and K. Kawasaki, J. Phys.: Condens. Matter 7, 4351 (1995)

-300 -150 0 150 300

(kHz)

S(k

,

propagating

overdamped

Tc=45o C @ k=24 mm-1

k

S(k,)

,Imk2

, 1B kDkT

kS

propagating

overdamped

The calculated Tc seems now to be correct.Can we use JK for full shape analysis of our XPCS data ?

T=const

k=const

Fluctuation-Dissipation

Analytical approximations are inapplicable in the critical damping regime !

10-6 10-5 10-4 10-3

G()

(sec)

Heterodyne vs. homodyne

10-6 10-5 10-4 10-3

(sec)

G

)

Beating against a reference beam

J.C. Earnshaw (1987)

Gra

ting

laser

PMT

bgdeqStItIG ti

t

2

),(~~

t

tEtEG ~

spontaneousswitching

Homodyne

Who ordered a reference beam for XPCS ?

Small effective sample size yields non-zero scattered intensity even for a perfectly smooth surface.

The effective sample size changes in time:

meniscus, dust particles, etc.

The scattering peaks move in time, sometimes interfering with the diffuse signal.

Why is the switching time-dependent ?

1 Gutt et al., PRL 91, 076104 (2005)2 Ghaderi, PhD thesis (2006)

What is the origin of the spontaneous reference beam in surface XPCS ?

Interference with the reflected beam, R(qz)(qx)(qy) ???

Fresnel (near field) conditions mix the low q values1.

Instrumental q resolution2.

Partial coherence of the beam.

Set the reference beam intensity as a free fitting parameter !

bgqCII

qCII

IqG

srsr

r

2

22 ,1

,2

,

deqSqqC ti),(,

Full shape analysis

Ir – reference beam-“static” diffuse scattering ts tqII , Unknown

Jäckle & Kawasaki (1995)Detector resolution

10-6 10-5 10-4 10-3

qx=78 mm-1

qx=56 mm-1

qx=36 mm-1

qx=26 mm-1

qx=13 mm-1

(s)

T=343 K

G(q

x,)

G(q

x,)

10-5 10-4 10-3

T=407 K

T=353 K

T=333 K

T=323 K

T=313 K

(s) (s)

qx=24 mm-1

L

2.5 3.0 3.5 4.0 4.5

0

2

4

371 K 303 K 343 K

ln(I

s)

ln(qx)

qx-2

Full shape analysis with only Is and Ir being free.

The fitted Is values show the diffuse intensity scaling known from CWT (for small qz values).

Let’s calculate the experimental S(q,w) spectra…

A. Madsen et al., PRL 92, 096104 (2004)

10-6 10-5 10-4 10-3

qx=78 mm-1

qx=56 mm-1

qx=36 mm-1

qx=26 mm-1

qx=13 mm-1

(s)

T=343 K

G(q

x,)F..{}

t

Stokeskin

qrip

kout

anti-Stokeskin kout

qrip

Let’s introduce surface viscoelasticity into the same formalism !

The spectra are fully described by the JK theory, assuming an unstructured interface. No need to introduce Freyland’s modified elastic term in the CW spectrum. No evidence for “ferroelectric” transition, at least for T<130 oC. Does it mean that the surface of an IL is not layered ?

-300 -150 0 150 300

78 mm-1

36 mm-1

17 mm-1

56 mm-1

78 mm-1

2

smooth

S(q

,)

(kHz)

T=343 K

Use the fitted values of Is and Ir to evaluate the experimental S(q,w)

-100 -50 0 50 100

(kHz)

S(q

,)

313 K

323 K

333 K

343 K

371 K

403 K 40 50 600

5

10

15

XPCS

Lucassen

Jäckle

24 mm-1

T (oC)

(

kHz)

, XPCSTheory

ln(T-Tc)

ln(

)

|T-Tc|-1/2

36 mm-1

24 mm-1

-3 -2 -1 0 1 2 3 40

1

2

3

ln(T-Tc)

qx=24 mm-1

dd i

i

k

k

iikik x

v

x

vp

02

2

xzx(i) “elastic” interface

Dilational modulus

02

2

zzx

h

Boundary conditions at the interface:

Stress tensor:

Surface tension

(ii)

Change the boundary conditions to introduce elasticity of the surface layer

0 xz(i) ideal interface – lateral displacement of surface elementh – surface normal displacement

D. Langevin and M.A. Bouchiat, C.R. Acad. Sc. Paris 272, 1422 (1971)

(kHz) d/

S(q,

)

d=0

time = d

(Hz)

d/

Resonance with Marangoni waves

The changes are too smallto be detected by XPCS…

T=100 oCq=17 mm-1

(kH

z)S(q

,)

d

Future directions Langmuir films on low-viscosity liquids ?

d

D

am

pin

g (

kHz)

19.0

19.5

20.0

0 1 2

2

3

4

(

kHz)

LF on water:qx = 20 mm-1

T = 25 oC

= 50 mN/m

Spontaneous surface structures:surface freezing in alkanes,surface layering in liq. metals…

Higher coherent flux and stable experimental conditions are needed to avoid spontaneous homo-hetero switching.

If switching is unavoidable, try to measure the intensity of the reference beam !

Go to higher q-values… (Far future: check dynamics at sub-micron scales)

S(q

,)/

Sm

ax

(kHz)16 18 20 22 24

0.0

0.5

1.0

d=0d>>

d=res

What have we learned about surface XPCS ?• Quantitative analysis, accounting for

both homo- and hetero-dyne terms,critical damping conditions (no Lorentizan app.),finite detector resolution

perfectly fits the experiment.

• Homo-hetero switching costs extra fitting parameters.

• Surface viscoelasticity is only measurable at low viscosity.

What have we learned about surface capillary waves ?

• Linear response theory (JK) provides the correct description of CW dynamics in an IL.

• Hydrodynamics alone (LLR) is insufficient.

• Critical scaling of CW frequencies at T → Tc , even though the hydrodynamic Tc is not the actual Tc .

Summary

Thanks …

Organizers (NSLS-II).

Audience.

Prof. Moshe Deutsch (Bar-Ilan University, Israel).

Dr. Ben Ocko (Brookhaven, USA).

Dr. Anders Madsen (ESRF, France).

Dr. Patrick Huber and M. Wolff (Saarland University, Germany).

Dr. Michael Sprung (APS, USA).

Dr. Julian Baumert (BNL, deceased).

Chemada Ltd. (Israel).

German-Israeli Science Foundation, GIF (Israel).